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Two order parameters for the Kuramoto model on complex networks.

Soon-Hyung Yook1, Yup Kim1

  • 1Department of Physics and Research Institute for Basic Sciences, Kyung Hee University, Seoul 130-701, Korea.

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We studied two order parameters in the Kuramoto model

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Area of Science:

  • Complex systems
  • Statistical physics
  • Network science

Background:

  • The Kuramoto model is a key tool for studying synchronization in coupled oscillator systems.
  • Order parameters are crucial for identifying distinct phases in dynamical systems.
  • Understanding phase transitions in complex networks is an ongoing challenge.

Purpose of the Study:

  • To investigate the behavior of two distinct order parameters within the desynchronized phase of the Kuramoto model.
  • To evaluate the efficacy of these order parameters in distinguishing between synchronized and desynchronized states on complex networks of varying sizes (N).

Main Methods:

  • Exact mathematical derivation of the difference (Δ) between two order parameters on a star network.
  • Analysis of order parameter behavior in the desynchronized phase.
  • Development and numerical verification of an analytic conjecture regarding order parameter agreement for large network sizes (N→∞).

Main Results:

  • The two order parameters exhibit disagreement within the desynchronized phase of the Kuramoto model.
  • The network's hub structure significantly influences the behavior of the order parameters.
  • An analytic condition was conjectured for the agreement of the two order parameters as N approaches infinity.

Conclusions:

  • The choice of order parameter can be critical for accurately characterizing the desynchronized phase in the Kuramoto model.
  • Network topology, particularly the presence of hubs, plays a vital role in the collective dynamics and phase identification.
  • The findings provide a theoretical basis for selecting appropriate order parameters in network synchronization studies.